Silvering of Lens Formula: A Silvered Lens as One Mirror

The silvering of lens formula is $P = 2P_L + P_M$ : when one face of a thin lens is coated so that it reflects, the whole system acts as a single mirror whose power equals twice the power of the lens plus the power of the silvered surface treated as a mirror. The lens counts twice because light passes through it on the way in and again on the way out.

On this page powers are taken as positive for converging elements and negative for diverging ones. The equivalent focal length is then $F = \frac{1}{P}$ in size, and a positive P means the silvered lens behaves like a concave mirror. New Cartesian signs give every concave mirror a focal length below zero, so write the result as $F = -\frac{1}{P}$ when you substitute into the mirror formula. Silvered lenses are a favourite problem type in JEE and NEET practice sets on ray optics.

Quick facts

Formula
$P = 2P_L + P_M$
Lens power
$P_L = \frac{1}{f_L}$ , positive for a convex lens
Mirror power
$P_M = \frac{2}{R}$ for a concave silvered face, 0 for a flat one
Equivalent focal length
$|F| = \frac{1}{P}$ ; a positive P acts as a concave mirror
Unit
dioptre (D) for P, metre for F
Plano-convex, flat face silvered
$|F| = \frac{R}{2(n - 1)}$
Plano-convex, curved face silvered
$|F| = \frac{R}{2n}$

Why the lens is counted twice

Follow a ray into a silvered lens. It refracts at the front face and crosses the glass, which is the lens acting once. It reflects at the coated back face, which is the mirror acting once. It then crosses the glass in the opposite direction and refracts out through the front face, which is the lens acting a second time. Three thin elements in contact simply add their powers:

$P = P_L + P_M + P_L = 2P_L + P_M$

Quantities in the silvered lens formula
Symbol	Meaning	Unit
$P$	Power of the equivalent mirror	D
$P_L$	Power of the lens on its own, in air	D
$P_M$	Power of the silvered face used as a mirror	D
$F$	Focal length of the equivalent mirror	m
$n$	Refractive index of the glass	no unit
$R$	Radius of curvature of the curved face	m

Silvering of lens formula for the standard cases

For a plano-convex lens, the lens maker result gives $P_L = \frac{n - 1}{R}$ . What changes between cases is the mirror term.

Flat face silvered. A plane mirror has zero power, so $P = \frac{2(n - 1)}{R}$ and $|F| = \frac{R}{2(n - 1)}$ . With n = 1.5 this is simply R.
Curved face silvered. Seen from inside the glass, the curved face is a concave mirror of focal length R/2, so $P_M = \frac{2}{R}$ . Then $P = \frac{2(n - 1)}{R} + \frac{2}{R} = \frac{2n}{R}$ and $|F| = \frac{R}{2n}$ . With n = 1.5 this is R/3.
Equiconvex lens, one face silvered. Here $P_L = \frac{2(n - 1)}{R}$ and $P_M = \frac{2}{R}$ , so $P = \frac{4(n - 1) + 2}{R}$ . For n = 1.5, P = 4/R and |F| = R/4.

Silvering a concave lens works the same way with a negative P_L; a plano-concave lens with its flat face silvered acts as a convex (diverging) mirror.

Worked examples

Example 1: flat face silvered

One side of a glass lens (n = 1.5) is flat and the other is curved with R = 20 cm. The flat side is silvered. Find the focal length of the system.

Work in metres: R = 0.20 m.

$P_L = \frac{0.5}{0.20} = 2.5\ \text{D}, \qquad P_M = 0$

$P = 2(2.5) + 0 = 5.0\ \text{D} \Rightarrow |F| = 0.20\ \text{m}$

The system acts as a concave mirror of focal length 20 cm, so F = −20 cm in the New Cartesian convention.

Example 2: curved face silvered

The same lens is silvered on its curved face instead. Find the new focal length.

$P_M = \frac{2}{R} = \frac{2}{0.20} = 10\ \text{D}$

$P = 2(2.5) + 10 = 15\ \text{D} \Rightarrow |F| = \frac{1}{15} = 0.0667\ \text{m} \approx 6.7\ \text{cm}$

That matches R/(2n) = 20/3 ≈ 6.7 cm.

Example 3: finding the image

An object is placed 40 cm in front of the silvered lens from Example 1. Where is the image?

Treat it as a concave mirror with F = −20 cm and u = −40 cm.

$\frac{1}{v} = -\frac{1}{20} + \frac{1}{40} = \frac{-2 + 1}{40} = -\frac{1}{40}$

v = −40 cm: the object sits at the centre of curvature of the equivalent mirror, so the real, inverted image lands on top of it with m = −1.

The calculator on this page works in dioptres with converging powers positive. Enter 0 for a plane silvered face.

Common errors with silvered lenses

Counting the lens once. Light crosses it twice, so the factor 2 is essential.
Using the radius of the wrong face for the mirror term. Only the silvered face acts as the mirror.
Forgetting that a flat silvered face still contributes: it contributes zero power, but it is what sends the light back through the lens.
Using the lens formula for the image. The system is a mirror, so use the mirror formula and m = −v/u.
Mixing sign conventions. Do the power sum with converging positive, then convert to a New Cartesian F at the end.

The lens power part comes straight from the power of lens formula and the lens maker formula. Image size uses the magnification formula of mirror. All of these are linked from the waves and optics hub.

A quick check by thinking about the light

The flat face silvered case is easy to picture. Parallel light enters the curved face and starts to converge. The flat mirror folds it back, so the light passes through the same lens again and converges twice as strongly. Doubling the power halves the focal length: a lens of focal length 2R (n = 1.5) becomes a mirror of focal length R. If your answer for a silvered convex lens is longer than the lens focal length on its own, you have made a sign slip.

Frequently asked questions

What is the silvering of lens formula?

Add the lens power twice to the mirror power: P = 2P(lens) + P(mirror), with the silvered surface treated as the mirror. The system acts as a mirror of focal length 1/P.

Why is the lens power doubled in a silvered lens?

Light passes through the lens twice, once going in and once coming back out after reflection. Each pass adds the lens power once.

What is the focal length of a plano-convex lens silvered on its flat side?

It is R/(2(n − 1)), with R taken from the curved side. For glass with n = 1.5, that equals R.

Does a silvered convex lens act as a concave or convex mirror?

It acts as a converging, or concave, mirror, because both the lens and a concave silvered face add positive power.

What is the power of a plane mirror in this formula?

Zero. With no curvature there is no focusing, so the mirror term is zero and only the doubled lens power remains.